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I'm reading a proof in Eymard's paper about the Fourier Algebra in which he refer to a proposition and a theorem in Roger Godement paper Les Fonctions De Type Positif et la Theorie Des Groupes, which can be found here (just click on view pdf) . is there an english translation for the hole paper?. if not, I'm trying to understand Proposition 12 in page 68 and Theorem 17 in page 73.

Any help would be appreciated.

Thank you.

Transcription of the paragraphs in question:

Page 68: Proposition 12. Toute fonction continue de type positif, sommable pour $dx/(\rho(x))^{1/2}$, est de carré sommable pour $dx$ (c'est à dire: $\mathcal P^1 \subset \mathcal P^2$).

Page 73: Théoreme 17. Toute fonction continue $\phi(x)$, de type positive et de carré sommable, est de la forme $$\phi(x) = \psi * \psi^\sim(x) = \int \psi(xy)\overline{\psi(y)}dy \quad \text{ où }\quad \psi \in \mathcal P^2.$$

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I can translate the French, though I'm unfamiliar with the math, so it'll be a very literal translation that hopefully you can interpret:

Prop 12: Every continuous function of positive type that is summable for $dx/(\rho(x))^{1/2}$, is square-summable for $dx$ (that is: $\mathcal{P}^1 \subset \mathcal{P}^2$).

If that means something to you, great! If not, then we'll need a French speaker who's a better mathematician that I to intervene!

Edit given addition to question:

Theorem 17: Every continuous function $\phi(x)$ that is positive and square-summable is of the form...

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  • $\begingroup$ No problem. Does it make sense to you? I hope so! $\endgroup$
    – Shane
    Nov 5, 2014 at 21:04
  • $\begingroup$ I'd say about the same, but I am as well not fluet in 'math-french'. But I think if the author wanted to say 'functions' wouldn't he have to write 'fonctions'? (plural form?) Thats just a detail that made me hesitate. But then again, it's math=P $\endgroup$
    – flawr
    Nov 5, 2014 at 21:04
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    $\begingroup$ Probably: “Every continuous function…” and “is square-summable…” $\endgroup$
    – Lubin
    Nov 5, 2014 at 21:07
  • $\begingroup$ Thanks Lubin. I was unfamiliar with that term, but you're right. $\endgroup$
    – Shane
    Nov 5, 2014 at 21:11
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    $\begingroup$ In Theorem 17, like in Proposition 12, the condition is that the function is “of positive type,” (not just “is positive”). The meaning of “of positive type” is defined at the top of page 23: A function of positive type is one that satisfies the condition (I, 12'') in the statement of Theorem 2 on page 21. In addition, the word “où” at the end of the expression in Theorem 17 means “where.” $\endgroup$
    – Steve Kass
    Nov 5, 2014 at 21:16

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